June 11, 2015

Slice-ribbon progress

There has been some recent interesting progress around the Slice Ribbon Conjecture. In particular, Yasui is giving talks on an infinite family counterexamples to the Akbulut-Kirby Conjecture (1978) that he has constructed:

Akbulut-Kirby Conjecture: If 0-surgeries on two knots give the same 3-manifold, then the knots with relevant orientations are concordant.

Note that some knots are not concordant to their reverses (Livingston), but the 0-surgery of a knot and its reverse are homeomorphic, so Akbulut-Kirby had to revise their original formalism to allow for arbitrary orientations. Abe and Tagami recently showed that if the Slice-Ribbon Conjecture is true then the Akbulut-Kirby Conjecture is false. Thus Yasui has eliminated an avenue to falsify the Slice-Ribbon Conjecture.